[EM] Strategies for RRV/RSV and BR for multi-member constituencies

[Kristofer Munsterhjelm]

Raph Frank wrote:
> On Fri, Jan 30, 2009 at 10:09 PM, Juho Laatu <juho4880 at yahoo.co.uk> wrote:
> PAV uses the rule
> 
> 1+1/3+1/5+1/7 + ... (i.e. terms = number of approved candidates elected)
> 
> However, it doesn't just average the results.  If one of your
> candidates are elected, then it counts as full strength, but the 2nd
> candidates counts at 1/3 of that strength.  The voter's happiness is 2
> units, but it only counts as 1.33 units.  Effectively, that voter's
> happiness is deweighted by 33%.
> 
> RRV doesn't quite work that way, but it gives the same kind of result
> to sequential PAV.
> 
> Anyway, maybe the system could be something like
> 
> For each possible winning set
> - work out the average utility for each voter of all the candidates in
> the winning set
> - sort the voters in order of their happiness
> - give each voter a weight dependent on the position in the ordering
> - the happiness for that result is equal to the average happiness
> using the above weightings
> 
> If the weighting was 1 no matter what, then it wouldn't be a PR method.
> 
> I wonder if there is a weighting that would achieve Droop proportionality.

I would guess that PAV, being based on a divisor method (Sainte-Laguë in 
the case above), must fail Droop proportionality to some extent, just 
like Webster's method must fail quota.

In reality, Webster's method fails quota very rarely. I'm not sure if 
PAV would fail it similarly rarely, but I doubt it, since PAV got quite 
bad scores in my simulator, and most methods that pass Droop 
Proportionality get significantly better scores; or maybe my 
simulation's way of setting approval cuttofs is bad.

Perhaps it would be possible to make PAV equivalents of apportionment 
systems that meet quota. I'm not sure how, though, and if the parallel 
between Webster and PAV doesn't hold, we can't directly port 
apportionment methods that way either.

> If a group of voters were to vote max for 1 candidate, and min for all
> the rest, I wonder is there a weighting function that will guarantee
> that that candidate will be in the best winning circle.

For PAV, that would be bullet voting. What do you mean by the "best 
winning circle" - the Smith set?